A private LUN is one that is used by the system and is not available directly to any host (you can't put it in a storage group). Some examples of private LUNs would be the LUN on a Global Hot Spare and an element of a MetaLUN.
So, that leads to the question of what exactly a MetaLUN is. In the simplest terms, a MetaLUN is a LUN that is made up of multiple "elements" (LUNs). The easiest example would be to think about a host that wants a single 200GB LUN allocated to it. You only have 100GB available on a single RAID group, so you look further and find a second RAID group with another 100GB free. Since you just want to allocate a single LUN to the host you build a MetaLUN by extending the first LUN with the second one. You end up with something that can be allocated as a single LUN but it looks like a single 200GB LUN to the host.
There are two kinds of MetaLUNs, and they are ideal for different applications.
Concatenated MetaLUNs are good for building a larger LUN from multiple smaller LUNs when you aren't so concerned about performance characteristics. It is just like a spanned volume on a Windows host... you just "tack" the new chunk of storage on the end of the existing LUN.
Striped MetaLUNs can allow you to build more performance potential into a LUN. Let's say you need a 200GB file system for a database. You know you are going to be driving very high IO through this file system... more than you can get out of your standard RAID5(4+1) RAID group. Instead of building a single LUN on a single RAID group you would build 4 x 50GB LUNs on four separate (but preferably identical) RAID groups. You then build a Striped MetaLUN with the four separate LUNs and end up with something that can potentially support 4 times the IO throughput.
This is a simplification, and there are lots of variables and best practices to consider... but this should get you started.
For your reading enjoyment on Powerlink..............
This white paper describes the LUN expansion capabilities available with EMC CLARiiON metaLUNs. It provides details on the terminology, concepts, OS support, storage-system operations, and management of metaLUNs.
MetaLUN: A Meta LUN is formed by combining more than one LUN with the configuration can be striped or Concatenated.
PrivateLUN: A LUN becomes private LUN when we add to Reserved LUN pool.